Asymmetric least squares smoothing

// Originally developed by tony withers:
//
// by tony.withers@uwo.ca, using method of Eilers, PHC and Boelens, HFM
// (2005) Baseline correction with asymmetric least squares smoothing.

// Creates (and overwrites) w_base, a baseline estimate for w_data. The
// asymmetry parameter (Eilers and Boelens' p) generally takes values
// between 0.001 and 0.1. Try varying lambda in orders of magnitude
// between 10^2 and 10^9. Not efficient for large N, try it for w_data
// with fewer than 1000 points.
//
//
// I just changed the code to avoid the slow matrix multiplication.
// The H-matrix is now constructed "manually". This saves time and memory
// allows larger datasets.
// (kmichel@wzw.tum.de)

function ALS(w_data, lambda, asymmetry)
    wave w_data
    variable lambda, asymmetry
    variable i, N=numpnts(w_data), rms=inf
    variable maxIts=20
   
    //    matrixOp /free  D = identity(N)
    //    differentiate /EP=1/METH=2/DIM=0 D
    //    differentiate /EP=1/METH=2/DIM=0 D

    //      this step (specifically the matrix multiplication) is slow:
    //      matrixOp /o H = lambda * (D^t x D)
   
    Make /O/N=(N) diag0 = 6
    wave diag0
    diag0[0]=1
    diag0[1]=5
    diag0[N-2]=5
    diag0[N-1]=1

    Make /O/N=(N-1) diag1 = -4
    wave diag1
    diag1[0]=-2
    diag1[N-2]=-2

    Make /O/N=(N-2) diag2 = 1
    Make /O/N=(N,N) H
    wave  H
    matrixoP/o H = setoffdiag(H,0,diag0)
    matrixop/o H = setoffdiag(H,-1,diag1)
    matrixop/o H = setoffdiag(H,1,diag1)
    matrixop/o H = setoffdiag(H,-2,diag2)
    matrixop/o H = setoffdiag(H,2,diag2)
    matrixop/o H = lambda * H
    killwaves diag0, diag1, diag2
    duplicate /o/free w_data w, w_new
    w=1

    for (i=0;i<maxIts;i+=1)
        matrixOp /o /free  C = chol(diagRC(w, N, N)+H)
        matrixOp /o w_base = backwardSub(C,(forwardSub(C^t, w * w_data)))
        w_new = asymmetry * (w_data>w_base) + (1-asymmetry) * (w_data<w_base)
       
        // convergence test
        w-=w_new
        wavestats /Q w
        if (v_rms>=rms)
            killwaves H    
            return i+1
        else
            rms=v_rms
            w=w_new
        endif
   
    endfor

    return 0
end